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The image contains a set of math problems. Let's go through each one briefly and outline their solutions:

1. **Find the derivative of \( f = 3t^4 - 2t^3 \)**
   - Differentiate each term of \( f(t) = 3t^4 - 2t^3 \) with respect to \( t \).

2. **Find the displacement \( x(t) \) of an object from \( A = 3t^2 \) at \( t = 5 \)**
   - Evaluate the position function \( A = 3t^2 \) at \( t = 5 \).

3. **Solve the equation \( \log(x + y) = \log 2 + \log(z) \)**
   - Use properties of logarithms to simplify and solve for \( x + y \).

4. **Calculate the surface area and volume of a hemisphere of diameter 15 cm**
   - Use the formulas for the surface area and volume of a hemisphere: 
     - Surface area \( A = 2\pi r^2 \)
     - Volume \( V = \frac{2}{3}\pi r^3 \)
     - With \( r = \frac{15}{2} \) cm.

5. **Find the inverse Laplace transform of \( F(s) = \frac{1}{s^2 + 4} \)**
   - Recognize the standard Laplace transform pair and find the corresponding time-domain function.

6. **Simplify the expression \( \log(\frac{e^3}{e^2}) \)**:
   - Apply logarithm rules to simplify.

7. **Solve the system of equations:**
   - \( 2x - 3y + z = 5 \)
   - \( 3x + y - z = 4 \)
   - \( 5x - 2y + 2z = 8 \)
   - Use methods like substitution, elimination, or matrix operations to solve for \( x \), \( y \), and \( z \).

8. **Find the coordinates of the vertex of the parabola \( y = 3x^2 - 18x + 5 \)**
   - Complete the square or use the vertex formula \( x = -\frac{b}{2a} \).

9. **Find the roots of the equation \( 4\cos^2 x - 3 = 0 \)**
   - Solve for \( \cos x \) and find the corresponding angles \( x \).

10. **Find the area of a triangle with vertices at \( A(1,2) \), \( B(3,-4) \), \( C(-1,2) \)**
    - Use the formula for the area of a triangle given by vertices coordinates.

11. **A box contains 4 red, 3 blue, and 2 green balls. Find the probability of drawing two red balls.**
    - Calculate the probability using the combination formula.

Would you like a detailed solution for a specific problem, or do you have any particular questions? Here are some follow-up questions related to the topics:

1. What is the importance of finding derivatives in real-world applications?
2. Can you explain the physical significance of displacement in mechanics?
3. How can logarithm properties simplify complex expressions?
4. What are some real-life examples where calculating surface area and volume of hemispheres is necessary?
5. How is the Laplace transform used in engineering and physics?
6. What are the key differences between natural logarithms and common logarithms?
7. What are some methods to solve a system of linear equations?
8. How does the vertex form of a parabola make it easier to graph?

**Tip:** When solving logarithmic equations, always remember to check for any extraneous solutions that might arise from the domain restrictions of the logarithms.

Explore solutions to mathematical problems involving derivatives, equations, logarithms, geometry, and more. Learn key formulas and theorems like the fundamental theorem of calculus and properties of logarithms. Suitable for high school students and above.

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